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296      XJV   INTEGRATION IN LOCALLY COMPACT GROUPS

But we have

(h(x)(g(x) - 0CT1*)) W(x) = (g(x)(h(x) - h(sx

J                                                                                          <J

and therefore,

/'

For each e > 0, it follows from (14.10.6.4) that there exists a compact neigh-
borhood V of e in G such that Nj(A  j(s~i)h) :g e for all s e V, and hence,

fn(s)dp(S)

JG

Choose TIO such that f   \fn(s)\ dfl(s)  s and   1 - ffn(s) dfi(s)

J (Jv                                                            J

< e for all

n^.n0; then we have

and

and therefore, finally

I (kyfn * 9 ~~ 9^ I   = (^ KA, ^) I   H- ^

for all  ^ w0, which shows that (/ * ^)~ converges weakly to ^.

Examples

(14.11.2) Let (Vn) be a fundamental system of neighborhoods of the neutral
element e in G. Since the support of /? is the whole of G, there exists for each
n a function fn e <#"+(G) with support contained in Vn and such that

\'fn(x)dp(x)^0 (13.19.1). Multiplying each fn by a suitable constant, we

may assume that j fn(x) dfi(x) = 1, and the sequence (/J now satisfies
the conditions of (14.11.1). We see therefore that every measure ju on G
may be approximated (in the sense of the vague topology) by a sequence
of "regularizations" which are measures having a continuous density with
respect to /J (14.9.3). \\fn(x)\ dfi(x) is bounded;